Convergence in Probability

Series: Stochastic Approximation from First Principles · Part 17 of 22 The energy proof controls entire sample paths. A more direct route expands $\Delta_k$ and studies: $$ \mathbb{E}[\Delta_k] \quad\text{and}\quad \operatorname{Var}(\Delta_k). $$If both approach zero, the distribution of $\Delta_k$ concentrates near zero. This route is intuitive, but it usually proves convergence in probability rather than the almost-sure convergence in Dvoretzky’s theorem. The Simplified Setting Begin with: $$ \Delta_{k+1} = (1-\alpha_k)\Delta_k + \beta_k\eta_k. $$For this route, treat $\alpha_k$ and $\beta_k$ as given coefficients and use ordinary assumptions such as zero-mean, bounded-variance, independent or uncorrelated disturbances. ...

Series: Stochastic Approximation from First Principles · Part 21 of 22 Suppose the expectation and variance calculations give: $$ \mathbb{E}[\Delta_k]\to0, \qquad \operatorname{Var}(\Delta_k)\to0. $$What kind of convergence follows? From the Second Moment to Probability By the variance identity: $$ \operatorname{Var}(\Delta_k) = \mathbb{E}[\Delta_k^2] - \left(\mathbb{E}[\Delta_k]\right)^2. $$Therefore: $$ \mathbb{E}[\Delta_k^2] = \operatorname{Var}(\Delta_k) + \left(\mathbb{E}[\Delta_k]\right)^2. $$Both terms approach zero, so: $$ \mathbb{E}[\Delta_k^2]\to0. $$For every $\varepsilon>0$, Chebyshev’s inequality gives: $$ \mathbb{P}(|\Delta_k|>\varepsilon) \le \frac{\mathbb{E}[\Delta_k^2]}{\varepsilon^2} \longrightarrow0. $$Hence: $$ \Delta_k\xrightarrow{p}0. $$This is convergence in probability. ...